Properties

Degree 16
Conductor $ 2^{40} \cdot 3^{24} \cdot 7^{8} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·13-s + 20·25-s + 72·37-s − 4·49-s − 64·61-s − 16·97-s − 72·109-s − 80·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 64·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 4.43·13-s + 4·25-s + 11.8·37-s − 4/7·49-s − 8.19·61-s − 1.62·97-s − 6.89·109-s − 7.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4.92·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{40} \cdot 3^{24} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6048} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 2^{40} \cdot 3^{24} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $2.995777697$
$L(\frac12)$  $\approx$  $2.995777697$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( ( 1 + T^{2} )^{4} \)
good5 \( ( 1 - 2 p T^{2} + 51 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \)
11 \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{4} \)
13 \( ( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \)
17 \( ( 1 - 26 T^{2} + 531 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{4} \)
23 \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \)
29 \( ( 1 - 16 T^{2} + 1362 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 80 T^{2} + 3138 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 - 9 T + p T^{2} )^{8} \)
41 \( ( 1 - 10 T^{2} + 2787 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 62 T^{2} + 4443 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + 70 T^{2} + 2187 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 44 T^{2} + 2646 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
59 \( ( 1 + 70 T^{2} + 5787 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + 16 T + 180 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{4} \)
67 \( ( 1 - 44 T^{2} + 3318 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
71 \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{4} \)
73 \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{4} \)
79 \( ( 1 - 110 T^{2} + 4923 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( ( 1 + 262 T^{2} + 30075 T^{4} + 262 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - 116 T^{2} + 5382 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.96356768244291460123816524908, −2.95279987940488461058959726773, −2.93986100037202406913883720451, −2.86338300770114560825383493215, −2.85217292964628235219593144575, −2.80200742545663678989121902775, −2.64110285411410659860404076209, −2.57860132009858808037705670405, −2.41956412658744092732424882135, −2.41809082446386980084621520154, −2.34035872810408318314180758878, −2.23799074721963651375769684095, −2.05211161776278149583071426150, −1.59713539203354763884373306138, −1.56488273980748129107658878565, −1.51267564013525909546234150124, −1.33541270929710765694681835280, −1.19241163964774588089922152663, −1.16258443914382381030516145491, −1.10649070392080508803323222944, −0.793123381242227450985761166101, −0.69583990142221778454823065342, −0.38409991216973001025554956948, −0.27461725918572087729231263113, −0.16263994893559228941018338420, 0.16263994893559228941018338420, 0.27461725918572087729231263113, 0.38409991216973001025554956948, 0.69583990142221778454823065342, 0.793123381242227450985761166101, 1.10649070392080508803323222944, 1.16258443914382381030516145491, 1.19241163964774588089922152663, 1.33541270929710765694681835280, 1.51267564013525909546234150124, 1.56488273980748129107658878565, 1.59713539203354763884373306138, 2.05211161776278149583071426150, 2.23799074721963651375769684095, 2.34035872810408318314180758878, 2.41809082446386980084621520154, 2.41956412658744092732424882135, 2.57860132009858808037705670405, 2.64110285411410659860404076209, 2.80200742545663678989121902775, 2.85217292964628235219593144575, 2.86338300770114560825383493215, 2.93986100037202406913883720451, 2.95279987940488461058959726773, 2.96356768244291460123816524908

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.